Dec 30, 2012

Coin Puzzle: Predict the Other's Coin

Problem:
Assume the following 3-player game consisting of several rounds. Players A and B build a team, they have one fair coin each, and may initially talk to each other. Before starting the first round, however, no more communication between them is allowed until the end of the game. (Imagine they are separated in different places without any communication infrastructure.)

A round of the game consists of the following steps:

(1) the team gives one dollar to player C.

(2) Both A and B toss their coins independently.

(3) Both A and B try to predict the other's coin by telling the guess to C. (No communication: A does not know the outcome of B's coin toss, and vice versa, nor the guess).

(4) If C verifies that both A and B guess the other's coin correctly, then C has to give 3 dollars back to the team.

C should play, it's a fun game. Also the expected value of their winnings is 25c, but C should be happy with the risk that they might lose. If the game can be played repeatedly C can expect to make a profit in the long run.

Let A be the event that A guesses correctly; B that B guesses correctly. P(A) = P(B) = 1/2

yes, even in the event that C is an honest player. A has 1/2 chances of being correct and B has 1/2 chances of being correct. this is a total of 1/4 chances of both being correct. Each round C gains 1 dollar minus 3 dollars multiplied by the chance of A and B being correct (1/4). so C leaves each round with 1 - (3/4) = 1/4. on average C should gain 25 cents each round (1/4 of a dollar). i've been out of school for a while am i missing something obvious?

Consider the following strategy. Both A and B always guess that the result of the coin toss of the other player is the same as theirs (so e.g. if A gets heads, they guess that B also got heads). With two independent coin flips there is 0.5 probability that the two coins are either both heads or both tails. In that case A and B would both guess correctly, so therefore they have a 0.5 chance of winning and 0.5 probability of losing.

So with probability 0.5 player C will earn 1 dollar and with probability 0.5 player C will lose 2 dollars (1 gained initially -3 given away in the end). Then the expected value of the game for C will be E = 0.5*1 + 0.5*(-2) = 0.5 - 1 = -0.5

Therefore if A and B are smart players (and decide on a good strategy like always guessing the same or always guessing the opposite of what they get) then C should NOT play this game.

(Note that if A and B would guess randomly each time, then the probability that they would get each others coin toss results correct would be 0.5 * 0.5 = 0.25 and the expected value for C would beE = 0.25*(-2) + 0.75*1 = -0.5 + 0.75 = 0.25,so then C SHOULD play the game)

C should not play this game.A and B can win with probability one half, simply by guessing that the other has the same toss (e.g., if A obtains tail, he guesses that B has tail as well, and vice-versa). Since both tosses are equal with probability one half, the expecte gain of C is1 - 3 * 1/2 = -1/2which is negative, so he should not play.

You din't take in account the fact that "A & B may initially talk to each other".This will allow them to make some strategy to guess correct answer with higher probability.e.g.Their strategy could be "They guess the same as their own coin toss."A (Guess by A) B (Guess by B) WinH H H H YesH H T T NoT T H H NoT T T T YesSo now this strategy increases their winning probability to 0.5so expected gain of C per round = 1*0.5-3*0.5 = -$1

Some of you are missing the point that "A and B can initially talk to each other"Some of you are calculating the payoff wrong. If C wins, it has 1 dollar. If C loses, it has to give 1-3=-2 dollars.

The strategy that A and B follow:A and B will just say the coin that they have. Hence, on HH and TT - they win. Otherwise, they lose. So, half probability C would win and half that C would lose. Since, C loses more when he loses but gains less when he wins, C should not play.

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I am an early stage technology investor at Nexus Venture Partners. Prior to this, I was a 3x product entrepreneur. Prior to this, I worked as a private equity analyst at Blackstone and as a quant analyst at Morgan Stanley. I graduated from Department of Computer Science and Engineering of IIT Bombay. I enjoy Economics, Dramatics, Mathematics, Computer Science and Business.

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